Monday, 30 July 2012

complex geometry - question about kahler cone of a compact kahler manifold

Hi to all!



I'm studying complex geometry from Huybrechts book "Complex Geometry"
and i have problems with an exercise, please can anyone help me?



I define the kahler cone of a compact kahler manifold X as the set



KXsubseteqH(1,1)(X)capH2(X,mathbbR)
of kahler classes. I have to prove that KX doesn't contain any line
of the form alpha+tbeta with alpha,betainH(1,1)(X)capH2(X,mathbbR)
and betaneq0 (i identify classes with representatives).



This is what i thought: i know that a form omegainH(1,1)(X)capH2(X,mathbbR)
that is positive definite (locally of the form fraci2sumi,jhij(x)dziwedgedoverlinezj and (hij(x)) is a positive definite hermitian matrix forallxinX) is the kahler form associated to a kahler structure. Supposing alpha a kahler class i want to show that there is a tinmathbbR such that alpha+tbeta is not a kahler class. Since betaneq0 i can find a tinmathbbR such that alpha+tbeta is not positive definite any more, now i want to prove that there is no form omegainH(1,1)(X)capH2(X,mathbbR) such that omega=dlambda with lambda a real 1-form and omega=overlinepartialmu with mu a complex (1,0)-form (what i'd like to prove is: correcting representatives of cohomology classes with an exact form i don't get a kahler class). From partialoverlinepartial-lemma and a little work i know that omega=ipartialoverlinepartialf with f a real function. And now (and here i can't go on) i want to prove that i can't have a function f such that alpha+tbeta+ipartialoverlinepartialf is positive definite.



Please, if i made mistakes, or you know how to go on, or another way to solve this, tell me.



Thank you in advance.

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